Let G=< a> be a finite cyclic group of order n. Suppose that n = dq where d and q are non-negative integers. Show that is the unique cyclic subgroup of order d in G

Q: Let G be an abelian group and let H= {g EG: g^n= e for some nEN}. Prove that H is a subgroup of G.

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Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: , Let G be a finite Abelian group of order n. Suppose G = {91,92,.. {91, 92, 9n} and let x be an…

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Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: Show that the Euclidean group E(n) is a group under function composition.

A: The Euclidean group E(n) consists of all the rigid transformations of n-dimensional Euclidean space.…

Q: 1. Let G be a cyclic group of order n and k❘n. Show that G has a subgroup of order k; "

A: To prove that a cyclic group ( G ) of order ( n ) has a subgroup of order ( k ) for every ( k ) that…

Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: Calculate the cyclic subgroup (15) < (Z24, +21)

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Q: Let G be a group of order p", where p is a prime and n is a positive integer. Show that G contains a…

A: Given :-  Let G be a group of order pn , where p is a prime and n is a positive integer.  To…

Q: b) Let G be a group of order 2p, where p is prime number. Prove that every proper subgroup of G is…

A: (b) G is a group of order 2p.        Now, we know, order of any subgroup of a group must divide the…

Q: (b) Find all bEG with order /b| = 7 bEG. are: (c) Construct in Ga subgroup H with index.(H) = 7

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Q: Construct a subgroup lattice for the group Z/48Z.

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Q: Subgroup of G. (b) Let G be an abelian group. Let T = {ge G: g = e for some positive integer n}.…

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

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Q: Let G be a group with IG|=209 then every proper subgroup of G is: * O Non abelian O Non cyclic O…

A: See the detailed solution below.

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

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Q: (1) Let G= be a finite cyclic group of order n. (a) Suppose that n = dq where d and q are…

A: Cyclic group: A group is said to be cyclic if it is generated by a element a i.e.,…